More Pentagonal Tilings

Here is the pentagonal tesselation that appeared on the first
page of this section with one
difference. Although it seemed the diamond piece cannot be avoided, the
"boat", with its lack of symmetry, was a piece I disliked. So,
after producing the tesselation, I changed and adjusted some
pieces so that every "boat" could be removed, by using decagons,
which do have fivefold symmetry, in the tesselation.

Note that while I was able to adjust out the boat pieces
in this entire diagram, as this was done on an ad hoc
basis, rather than by modifying the pieces used in the underlying
recurrence relation, it is not a proof that it is possible to
always eliminate the boat piece completely. But such a proof, of
course, is provided in the page about Penrose tilings, where the
pieces used in the drawing above were used to build up tiles that could
be used to tile the entire plane as the components of a Penrose kite and
dart tiling.

The starting point for the tiling above is a pentagonal tiling that
looks very much like the original Penrose tiling:

Unlike that tiling, however, the diamonds, when enlarged, are oriented
with the star towards the pentagon from which they arose rather than with
the boat piece in that direction. Again, this was to delay the use of the
less-symmetric boat piece as much as possible. Then, the tiling on the
first page was developed by adding double-sized pentagons to the twice-enlarged
boat and star pieces, to somewhat reduce the need for boat pieces.

Then, the first step towards modifying that basic tiling was to
reduce the number of hat pieces needed by introducing double-sized
pentagons, as shown below:

That led to the decorative tiling on the first page, the double-sized
pentagons adding to the variety of the pattern. To eliminate the rest of
the hat pieces, decagons were introduced, also as shown in the diagram, but
although that led to the hat pieces being completely eliminated, it also
meant the decagons tended to dominate the arrangement, making it less
decorative from my own aesthetic point of view.

This tiling may not have the mathematical elegance of the Penrose tiling,
but it certainly has decorative possibilities.

One way to consider the available possibilities would be, by analogy with
the hexagon-boat-star tiling for the basic shapes, to consider these shapes,
used to supplement a pentagonal tiling, as being associated with a hexagon,
a boat, a decagon, and a fat star, as illustrated in the diagram at left.

The boat-only tiling on the previous page, suggestive of fish scales,
leads us to ask if we can do without the hexagon, so as to make a tiling
from pieces with five-fold symmetry only. Clearly, we would have to exclude
the double-sized pentagon to do so; while two adjacent decagons can adjoin
one of the points of the fat star, then a hexagon would be needed.

In the case of the star, it is not quite as obvious; five decagons
could surround a star. But over a longer range, the tiling would have to
be like a tiling with stars and pentagons only, which is not possible.

So, let us perhaps content ourselves with attempting to find a systematic
method for tiling the plane with the hexagon, star, decagon, and fat star
other than the one already obtained based on the Penrose kite and dart
tiling.

Of course, it certainly is possible to tile the plane with pentagons
only if the pentagons are not regular pentagons:

In this well-known example, the symmetry is of a conventional type,
with a repeating cell. This particular tiling is sometimes called the
"Cairo tiling", because it was noted as being used there in street paving. Trying
to begin with three irregular pentagons that begin at a point,

I obtain a tiling with a few hexagons as well. But, trying harder,
I finally do attain a tiling with only irregular pentagons and a hexagonal
cell; however, no overall reduction in the amount of distortion in the
shapes of the pentagons over that in the Cairo tiling is achieved:

Of course, if one is going
to look at conventional tilings, one might as well note the 17 possible
symmetry groups available for such tilings,

and then, having exhausted the subject (well, not quite, hardly
having even scratched the surface of the beauty possible from conventional
tilings), hastily depart it.